Christine currently has $5000 in her 401(k) and plans to contribute $100 each month for the next 30 years into it. What will be the value of Christine's 402(k) in 30 years if the per annum rate of return is assumed to be 12% compounded monthly?

To calculate the future value of Christine's 401(k) in 30 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment
P = the initial principal amount ($5000)
r = the annual interest rate (12% or 0.12)
n = the number of times that interest is compounded per year (monthly, so 12)
t = the number of years (30)

Plugging in the values:

A = 5000(1 + 0.12/12)^(12*30)

First, simplify the exponent:

A = 5000(1 + 0.01)^(12*30)

Next, calculate the compound interest within the parentheses:

A = 5000(1.01)^(12*30)

Now, calculate the value of (1.01)^(12*30):

A = 5000 * (1.01)^360

Using a calculator, raise 1.01 to the power of 360:

A ≈ 5000 * 12.17093

Calculating the final result:

A ≈ $60,854.65

Therefore, the value of Christine's 401(k) in 30 years would be approximately $60,854.65.